‘Trillion triangles’ advances ancient mathematics problem

It’s one of the oldest unsolved problems in mathematics, one that has intrigued people for centuries in part because it is deceptively simple. The question asks which whole numbers can express the area of a right-angled triangle with sides that are whole numbers or fractions. First posed more than a thousand years ago by a Persian mathematician, scholars around the world have puzzled over it since.

One of the most recent to tackle the problem is Robert Bradshaw, a Woodinville native and UW doctoral student in mathematics. This fall Bradshaw was part of the international team that announced it had found all such numbers up to a trillion, a major advance that discovered more than 3 billion new numbers that fit this description.

Their announcement has attracted attention from both mathematical experts and the broader public.

“There’s been a fair bit of interest. It’s the kind of thing an elementary or high school math teacher can tell his or her class about,” Bradshaw said.

Actually, Bradshaw belonged to one of two teams that made the announcement. He and collaborators Mark Watkins at the University of Sydney in Australia and David Harvey of Courant Institute at New York University formed one team. Their friendly competition was made up of researchers in England and Uruguay who had proposed renewing the search for such numbers.

“They had been talking about it for a while,” Bradshaw said. At an international mathematics conference in Spain last year, another mathematician challenged the group to revisit this ancient problem with a trillion as the goal, “and we said, ‘we could just do it,'” Bradshaw said. “That kind of spurred [the other group] on, got them working on it again.”

The two groups worked independently but came up with the same answer. They announced the results jointly in late September and are now working together on a formal write-up of the findings.

Many great mathematicians have been interested in congruent numbers — numbers that are also the area of a right-angled triangle whose three sides are whole numbers or fractions. For example, a right-angled triangle with sides measuring 3, 4 and 5 has an area of 6, so 6 is a congruent number. And a right-angled triangle with sides measuring 3/2, 20/3 and 41/6 has an area of 5, so 5 is a congruent number. But as the numbers increase the calculations get more difficult.

The new project looks for congruent numbers up to a trillion — that’s 1 with twelve zeros after it, or one million million. Both teams used computers to tackle the problem by reducing it to multiplying huge numbers. It’s a fairly simple concept in theory, Bradshaw said, except that each of the numbers to be multiplied was too big to fit in a computer’s memory. In fact, if the digits were written out by hand they would stretch to the moon and back.

So Bradshaw and his colleagues wrote a code that would split up the numbers and do the multiplication in pieces. They wrote the code over two weeks in July 2008 at a UW workshop. They then performed the calculations over two days on the UW’s Sage computer, using 128 gigabytes of accessible memory and 3 terabytes of storage space, to come up with the result.

“This is the oldest unsolved problem in mathematics,” said William Stein, associate professor in the Department of Mathematics and Bradshaw’s thesis adviser. “People have thought about it for a long time and have remained in a sense stumped as to how to solve it.”

“This pushed things pretty far beyond what people had done before,” which may help uncover a universal solution, Stein said. “Understanding this problem could be exactly what’s needed to understand many other interesting and important questions in mathematics.”

Bradshaw will graduate later this year and has already been hired as an engineer in Google’s Seattle office.

Bradshaw’s doctoral research concerns related topics in number theory, the branch of pure mathematics that studies properties of numbers. Although he didn’t know much about the history of the problem before taking up the “trillion triangle” challenge, his research touches on many of the same themes.

He cautions that the triangle results rely on the unproven Birch and Swinnerton-Dyer Conjecture, proposed in the early 1960s and, Bradshaw said, widely believed to be true. Proving that conjecture, however, is one of seven Millennium Prize Problems posed by the Clay Math Institute, which has put up a $1 million prize for a successful proof.

Bradshaw’s research concerns questions related to or inspired by the conjecture.

“I didn’t make proving it my thesis topic,” he said, “because who knows how long I’d be in grad school if I did that.”